
# q-expansion of newform 1815.2.a.m, downloaded from the LMFDB on 25 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 1815
weight = 2
poly_data = [1, -3, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0], 1], [[-2, 0, 1], 1], [[2, 2, -1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 1, 0], [1, 0, 0], [1, 0, 0], [0, -1, 1], [0, 0, 0], [0, 1, 1], [2, -3, -1], [-2, 0, -2], [0, -2, 2], [4, -2, 0], [4, -2, -2], [-2, 0, 0], [4, 2, 0], [0, -1, -3], [-4, 2, 2], [-2, 2, -2], [4, 2, -2], [2, 2, -2], [0, -2, -2], [4, -2, -2], [4, 3, -1], [-6, 4, 2], [-2, 3, 3], [-2, -4, 0], [6, 2, 2], [8, 4, -2], [-4, 4, 0], [-2, -1, -1], [2, -4, 4], [-6, 0, 0], [12, -3, -1], [0, -2, -2], [2, -4, -4], [-14, 2, 0], [4, 4, 2], [2, -8, -2], [2, -6, 2], [0, 0, -4], [-2, -3, 5], [6, -3, -1], [0, 8, -4], [-6, 4, 0], [-4, 0, 4], [0, 3, -5], [6, -7, -5], [4, 0, 4], [-2, -4, 2], [0, -2, 2], [-14, 1, 1], [14, -4, -4], [10, -1, -3], [-8, 2, 6], [-10, 2, 6], [24, -2, -2], [2, 2, -2], [10, -1, 7], [6, 4, 4], [-2, 4, -2], [0, 9, 1], [4, 2, 0], [-24, 3, 1], [6, 3, 1], [20, -1, 1], [16, -10, -6], [2, 4, -4], [14, 4, 4], [4, 2, 6], [-20, 3, -1], [-2, -1, -1], [-14, -2, 2], [2, -2, -6], [-12, 2, 2], [-8, -14, 6], [-8, 7, -5], [-20, 0, 0], [-12, 12, 8], [2, 4, 0], [-14, 2, 2], [2, 0, 0], [2, -8, -4], [8, -4, 0], [18, 0, -4], [-4, 0, -12], [-2, -6, 2], [18, -6, -4], [-8, 8, 4], [-2, 0, -12], [0, -11, 1], [-4, -4, 10], [-8, -2, -6], [12, 2, -2], [16, -2, 2], [-28, -6, 2], [-24, -4, 0], [0, 8, -4], [-2, 5, -3], [-2, 16, 0], [-14, 8, 0], [-20, -3, 3], [18, 0, 0], [-4, 13, -5], [10, -7, -1], [-18, 17, 5], [20, 4, -2], [-10, 14, 0], [-2, 10, 10], [28, -6, -2], [-26, 5, 3], [-16, 8, 0], [10, -2, -2], [12, -3, -1], [-32, -1, -5], [-6, 4, 4], [-4, -14, -2], [-12, -4, 8], [-2, -4, 8], [8, 0, 4], [-4, 12, -8], [6, -10, 2], [28, 4, -4], [-10, 4, -4], [8, 3, 3], [-14, -9, 1], [0, -10, 6], [-4, 8, 0], [16, -2, 4], [10, 8, -4], [-4, -4, 8], [0, 12, -4], [28, 3, 3], [14, 10, 4], [6, 3, -9], [-4, 12, 8], [10, -2, -2], [12, 0, -6], [10, -2, 6], [38, 2, -2], [-16, -13, 1], [14, -20, -4], [24, -2, 0], [2, -6, -8], [28, 6, -8], [4, -20, 0], [-10, -1, -9], [2, -8, -4], [-8, 2, 6], [-16, -13, 7], [-14, 17, 3], [12, 18, -2], [-4, 8, -12], [0, -9, -5], [-10, 8, 4], [-20, 16, 8], [-30, 5, -7], [-4, 2, -18], [12, -4, -8], [-14, 22, 0], [-10, -8, 4], [12, -7, 1], [-8, -12, -10], [-8, -6, 2], [22, 3, -3], [-12, 5, -9], [-4, 12, -12], [-6, 16, 8], [-4, -8, -4], [-12, -2, -18], [16, -17, -5], [-2, -18, 2], [22, -2, -6], [12, 12, -4], [-22, 24, 4], [-12, -2, 6], [24, -7, -11], [-8, 14, 10], [-10, 8, -4], [-26, -6, 4], [30, -12, 4], [-4, 1, -13], [6, 0, -4], [-8, -4, -4], [-20, 8, 8], [-22, -6, 2], [38, 3, -3], [-44, -2, -2], [18, 4, 16], [-12, 11, 3], [20, -2, 2], [-6, 4, -8], [-12, -20, 0], [-2, -2, -10], [6, -17, -9], [8, -12, -8], [-6, 4, 0], [22, 1, -3], [-6, -2, -14], [10, -6, 10], [-10, 4, 4], [-10, 7, 17], [-10, 7, 3], [32, 10, -8], [-34, 10, 8], [-26, -4, -12], [6, 6, 18], [-16, 4, -8], [-2, -16, 8], [24, 8, 0], [22, -11, 9], [8, 2, 12], [16, -6, 2], [-52, -3, 5], [-6, 12, 16], [0, 4, 4], [-4, -14, -10], [-4, -10, 6], [26, 4, 12], [12, -5, -15], [0, -8, 6], [-8, -12, -20], [-26, -6, -2], [-42, 10, 10], [-22, 12, 2], [38, 8, 4], [-20, 12, 0], [-6, 11, -1], [-10, 8, -4], [-6, 12, 4], [16, -6, 14], [-40, -7, -1], [28, -18, -6], [-54, -2, -2], [2, -18, -4], [-2, 18, 16], [0, 10, 12], [-24, 10, 2], [-2, -9, 3], [-34, 24, 4], [-10, -9, -19], [8, 10, 18], [-32, -8, 0], [-8, 4, 16], [2, -16, 2], [24, -12, -4], [34, -24, -4], [-22, -1, 5], [-28, -18, 6], [4, 10, 26], [24, 6, 14], [-18, -2, 8], [-18, 3, 7], [40, 9, 1], [0, 16, 14], [0, 12, -4], [10, -12, -4], [10, 1, -9], [24, -24, -4], [42, 8, -4], [-16, 5, 7], [22, -4, -4], [-20, -3, -3], [-4, -25, 5], [-18, -3, 9], [22, -6, 10], [20, -19, 1], [26, -18, 2], [-32, -8, 12], [-6, -4, -24], [-10, 24, -4], [-36, 23, 9], [-14, -9, 17], [-2, -12, 12], [-36, -6, 14], [-34, 22, 6], [-54, -2, -8], [-16, 9, 5], [4, -6, 18], [24, 6, -2], [10, -18, -6], [10, -26, -10], [-16, 2, 10], [2, -9, -17], [-20, -6, -14], [-10, 7, -13], [30, -4, -16], [-4, -3, 3], [-32, -24, 8], [30, 2, 10], [-38, 7, 9], [-20, -10, -10], [24, -8, -2], [2, -20, 0], [8, 0, 20], [-14, 3, 1], [24, -10, 6], [-12, 21, 9], [-28, 0, -18], [-44, 16, -4], [-2, -26, 2], [-16, 24, 12], [4, -33, 1], [12, 5, 21], [-30, 9, -9], [-18, 10, -8], [-28, 10, -2], [-24, -10, 14], [-14, 0, 8], [-24, -10, -2], [6, 4, 4], [48, 0, 8], [16, 3, -9], [-26, 5, 13], [18, 4, 16], [0, 24, -10], [8, 10, -6], [-10, -15, -7], [22, 6, -6], [-20, 4, -12], [28, -14, 2], [-18, -22, -6], [12, 20, -18], [-10, -16, -10], [54, 6, -2], [40, -6, 0], [28, -18, -2], [26, -1, 13], [38, -16, 4], [0, -26, -2], [-20, -4, -12], [18, 3, -13], [-18, 9, 19], [6, -2, 14], [6, 0, 8], [2, -20, 2], [-30, 1, 1], [10, 8, 18], [-16, 28, -8], [-18, -4, 4], [-22, 7, -19], [46, 0, 4], [-40, -1, 1], [42, -10, -2], [-54, -2, -6], [-12, 16, -10], [-40, -6, 6], [-34, 18, -2], [36, 14, -6], [54, 8, 8], [0, 8, 28], [-8, 8, 0], [6, -18, 2], [10, 22, -4], [10, 8, 8], [18, 12, 0], [4, -3, 7], [-38, -14, 14], [10, -3, -17], [52, -6, 10], [12, 10, 14], [-2, -21, -11], [-24, 20, 12], [44, -27, -7], [8, 4, -22], [-16, 2, -10], [4, -10, -6], [-4, 20, -12], [-2, 18, 10], [-22, -15, -9], [16, -5, 5], [-54, -2, 14], [-36, 18, -2], [-52, -18, 2], [46, -9, -13], [40, 4, -2], [-30, -2, 8], [34, 2, 10], [-32, -8, 12], [40, -10, 2], [-20, 1, 1], [12, 8, -22], [0, 33, -11], [-6, -20, -8], [-22, -10, -6], [52, 9, 3], [26, 5, -1], [-6, 28, -10], [-32, -4, -4], [-12, 12, 0], [-30, -2, -2], [-20, 21, 11], [-48, 16, 16], [-34, -12, -8], [30, 18, -18], [-16, -14, -14], [0, -10, 22], [8, 2, -10], [-24, -19, 17], [-10, 6, -28], [18, -12, 4], [-12, 16, 8], [-28, 18, 20], [30, -12, 16], [50, -20, -4], [-64, 15, 9], [-22, 10, -10], [12, -8, 6], [18, 24, -2], [10, -10, 22], [-16, 4, -6], [8, -14, -14], [24, -22, 10], [60, -23, -7], [-2, 3, -23], [68, -4, -4], [-10, -2, 8], [-20, -1, 11], [-10, -24, 0], [-20, -4, -8], [20, -4, -16], [-6, -6, -26], [38, 9, -15], [34, -16, 12], [-20, 17, -3], [32, -20, 4], [12, 14, 10], [34, -16, 8], [38, -18, -6], [-20, -20, -4], [8, -20, 18], [-28, 16, -8], [16, -18, 10]]
